with the next higher order, third order, dispersion [1]}, [2]}, [3]}, [4]}.
The Kahawara equation arises for surface water waves when the effect of surface tension is included and the NLS equation
with third order dispersion arises in nonlinear fibre optics, with third order dispersion being one of the higher order
effects included for femtosecond pulses [5]}. The effect of resonant radiation on the DSW structure will be
discussed in detail below, but it is sufficient to state here that if the effect of fifth order dispersion for the
Kahawara equation is large enough, \(\mu \ge 0\)  below a critical value, then the classical KdV DSW structure disappears
[6]}, [7]}, with a strong resonant wavetrain terminated by a Whitham shock, a modulation shock wave in the modulated wave
variables, rather than by a KdV-type DSW [7]}, [9]}, as illustrated in Figure REF (d). For DSW bearing
nonlinear, dispersive wave equations, the corresponding modulation equations are hyperbolic and so, in theory, possess sharp
gas dynamic type shock waves in the wave modulation parameters. When he originally derived modulation theory
[10]}, [11]}, [12]}, Whitham speculated on the applicability of these modulation shock solutions, which seem to
contradict the slowly varying assumption behind modulation theory. Recent work [9]}, [14]} has shown that these
modulation jump conditions are applicable and can be used to derive a wide variety of solutions for nonlinear, dispersive wave
equations which have been derived in the past using more involved methods, or not at all. Of interest here, they can be used
to derive non-KdV type DSW solutions for which there is resonance with radiation, which have been derived in the past using more
ad-hoc methods [6]}, [7]}.
